1. **Problem Statement:** Imagine a company wants to analyze the performance of two products over time. They collect data on sales and customer satisfaction, represented by a matrix $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$. The company wants to understand the underlying factors affecting these metrics by finding the eigenvalues, eigenvectors, and diagonalizing the matrix. 2. **What are Eigenvalues and Eigenvectors?** Eigenvalues are scalars $\lambda$ such that when the matrix $A$ acts on a vector $\mathbf{v}$ (the eigenvector), the output is just a scaled version of $\mathbf{v}$: $$A\mathbf{v} = \lambda \mathbf{v}$$ This helps identify directions in which the transformation acts by simply stretching or compressing. 3. **Finding Eigenvalues:** We solve the characteristic equation $$\det(A - \lambda I) = 0$$ where $I$ is the identity matrix. Calculate: $$\det\left(\begin{bmatrix}4 - \lambda & 1 \\ 2 & 3 - \lambda\end{bmatrix}\right) = (4 - \lambda)(3 - \lambda) - 2 \times 1 = 0$$ Expanding: $$ (4 - \lambda)(3 - \lambda) - 2 = 12 - 4\lambda - 3\lambda + \lambda^2 - 2 = 0 $$ Simplify: $$ \lambda^2 - 7\lambda + 10 = 0 $$ 4. **Solving the quadratic:** $$ \lambda = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm 3}{2} $$ So, $$ \lambda_1 = 5, \quad \lambda_2 = 2 $$ 5. **Finding Eigenvectors:** For $\lambda_1 = 5$, solve $(A - 5I)\mathbf{v} = 0$: $$ \begin{bmatrix}4 - 5 & 1 \\ 2 & 3 - 5\end{bmatrix} \mathbf{v} = \begin{bmatrix}-1 & 1 \\ 2 & -2\end{bmatrix} \mathbf{v} = 0 $$ From the first row: $$ -v_1 + v_2 = 0 \Rightarrow v_2 = v_1 $$ Eigenvector: $$ \mathbf{v}_1 = \begin{bmatrix}1 \\ 1\end{bmatrix} $$ For $\lambda_2 = 2$, solve $(A - 2I)\mathbf{v} = 0$: $$ \begin{bmatrix}2 & 1 \\ 2 & 1\end{bmatrix} \mathbf{v} = 0 $$ From the first row: $$ 2v_1 + v_2 = 0 \Rightarrow v_2 = -2v_1 $$ Eigenvector: $$ \mathbf{v}_2 = \begin{bmatrix}1 \\ -2\end{bmatrix} $$ 6. **Diagonalization:** Matrix $A$ can be diagonalized as: $$ A = PDP^{-1} $$ where $$ P = \begin{bmatrix}1 & 1 \\ 1 & -2\end{bmatrix}, \quad D = \begin{bmatrix}5 & 0 \\ 0 & 2\end{bmatrix} $$ 7. **Interpretation:** Diagonalizing $A$ simplifies understanding the system by decoupling the effects into independent components scaled by eigenvalues. **Final answer:** Eigenvalues: $5$ and $2$ Eigenvectors: $\begin{bmatrix}1 \\ 1\end{bmatrix}$ and $\begin{bmatrix}1 \\ -2\end{bmatrix}$ Diagonal matrix $D = \begin{bmatrix}5 & 0 \\ 0 & 2\end{bmatrix}$ and matrix $P = \begin{bmatrix}1 & 1 \\ 1 & -2\end{bmatrix}$ such that $A = PDP^{-1}$.